Subgroups — #34009709

Order 18688.

284 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626288	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556665	C₂	1	1	yes	maximal, characteristic
3	#22624846	C₂	1	1	yes	maximal, characteristic, complemented
4	#22624844	C₂	1	1	yes	maximal, characteristic, complemented
5	#22624843	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
7	M₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556665	C₄	1	2	yes	—
9	C₄⋊#11924031	C₄	1	2	yes	—
10	C₂²⋊#11924031	C₄	1	2	yes	—
11	#17989475	C₄	1	2	yes	—
12	C₄×#11924031	C₂²	1	1	yes	characteristic
13	C₄×#11924031	C₂²	1	1	yes	characteristic
14	C₄×#11924031	C₂²	1	1	yes	characteristic
15	C₄×#11924031	C₂²	1	1	yes	characteristic
16	#17988640	C₄	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	#17988308	C₄	1	1	yes	characteristic
19	#17988308	C₄	1	1	yes	characteristic
20	C₈×#10596831	C₂²	1	1	yes	characteristic
21	C₈×#10596831	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
23	#14556665	—	2	4	no	—
24	C₂×#11924031	C₂×C₄	1	2	yes	—
25	C₂×#11924031	C₂×C₄	1	2	yes	—
26	C₂×#11924031	C₂×C₄	1	2	yes	—
27	C₂×#11924031	C₂×C₄	1	2	yes	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₄⋊#10596831	C₈	1	2	yes	—
31	C₂²⋊#10596831	C₈	1	2	yes	—
32	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₄×#10596831	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	C₈	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#97×C₇₃	C₈	1	2	yes	—
41	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924031	—	2	2	no	—
44	#11924031	—	2	2	no	—
45	#11924031	—	2	2	no	—
46	#11924031	M₁₆	1	4	yes	complemented
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	2	2	no	—
49	C₂×#10596831	C₄²	1	1	yes	characteristic
50	C₂×#10596831	C₄²	1	1	yes	characteristic
51	C₂×#10596831	C₂×C₈	1	2	yes	—
52	C₂×#10596831	C₂×C₈	1	2	yes	—
53	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
54	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
58	M₁₆×C₇₃	—	2	4	no	—
59	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
61	C₄²×C₇₃	C₄²	1	1	yes	characteristic
62	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₈×C₇₃	—	2	2	no	—
77	C₈×C₇₃	M₃₂	1	4	yes	complemented
78	#10596831	—	2	1	no	—
79	#10596831	—	2	1	no	—
80	#10596831	—	2	2	no	—
81	#10596831	—	2	2	no	—
82	#10596831	C₂×M₁₆	1	2	yes	—
83	#10596831	#131	1	2	yes	—
84	C₂²×C₇₃	—	2	1	no	—
85	C₂²×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	—	2	1	no	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₇₃	#405	1	1	yes	characteristic
93	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
94	C₄×C₇₃	#504	1	2	yes	—
95	#11409	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	—	4	1	no	—
97	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
98	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
99	C₂×C₇₃	#2108	1	1	yes	characteristic
100	#2162	—	73	1	no	—
101	C₄×M₃₂	—	73	1	no	—
102	C₈×M₁₆	—	73	1	no	—
103	#1312	—	73	1	no	—
104	#1312	—	73	1	no	—
105	#1311	—	73	1	no	—
106	C₈×C₁₆	—	73	1	no	—
107	C₇₃	#11409	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₃₂	—	73	2	no	—
109	#431	—	73	1	no	—
110	C₄×M₁₆	#64408	1	1	yes	characteristic
111	C₂×C₄×C₈	—	73	1	no	—
112	C₄⋊C₁₆	—	73	2	no	—
113	C₂²⋊C₁₆	—	73	2	no	—
114	#346	—	73	2	no	—
115	C₄×C₁₆	—	73	1	no	—
116	C₄×C₁₆	—	73	1	no	—
117	C₄×C₁₆	—	73	1	no	—
118	C₄×C₁₆	—	73	1	no	—
119	#322	—	73	1	no	—
120	#322	—	73	1	no	—
121	C₈²	—	73	1	no	—
122	C₈²	—	73	1	no	—
123	C₂×M₁₆	#10596833	1	2	yes	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
127	M₃₂	—	146	4	no	—
128	C₂×C₁₆	—	73	2	no	—
129	C₂×C₁₆	—	73	2	no	—
130	C₂×C₁₆	—	73	2	no	—
131	C₂×C₁₆	—	73	2	no	—
132	C₄⋊C₈	—	73	2	no	—
133	C₂²⋊C₈	—	73	2	no	—
134	#97	#10596833	1	2	yes	—
135	C₄×C₈	—	73	1	no	—
136	C₄×C₈	—	73	1	no	—
137	C₄×C₈	—	73	1	no	—
138	C₄×C₈	—	73	1	no	—
139	C₄×C₈	—	73	1	no	—
140	C₄×C₈	—	73	1	no	—
141	C₄×C₈	C₂×#64408	1	1	yes	characteristic
142	C₄×C₈	C₂×#64408	1	1	yes	characteristic
143	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
144	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
145	C₂²×C₄	#11924036	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	73	1	no	—
148	C₂×C₈	—	73	1	no	—
149	C₂×C₈	—	73	2	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	73	1	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	146	2	no	—
154	C₂×C₈	—	146	2	no	—
155	C₂×C₈	C₂×#10596833	1	2	yes	—
156	C₂×C₈	C₂×#10596833	1	2	yes	—
157	C₄²	C₄×#64408	1	1	yes	characteristic
158	C₄²	C₂×#10596833	1	1	yes	characteristic
159	C₄²	#11924036	1	1	yes	characteristic
160	C₄²	C₂²×#64408	1	1	yes	characteristic
161	C₁₆	—	73	4	no	—
162	C₁₆	—	146	2	no	—
163	C₁₆	—	146	2	no	—
164	C₁₆	—	146	2	no	—
165	C₂³	C₄×#10596833	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
171	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
172	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
174	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
175	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	73	2	no	—
178	C₈	—	73	2	no	—
179	C₈	—	146	2	no	—
180	C₈	—	146	2	no	—
181	C₈	—	146	1	no	—
182	C₈	—	146	1	no	—
183	C₈	#14556665	1	4	yes	complemented
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	M₁₆×#64408	1	1	yes	characteristic
192	C₄	#17989553	1	1	yes	characteristic
193	C₄	C₂×#14556665	1	2	yes	—
194	C₄	#17989646	1	2	yes	—
195	C₂	—	4	1	no	—
196	C₂	M₁₆×#10596833	1	1	yes	characteristic
197	C₂	C₄×#14556665	1	1	yes	characteristic
198	C₂	#22626142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄×M₁₆×C₇₃	C₄
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#11409
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #22624846	C₂	semidirect
4. #22624844	C₂	semidirect
5. #22624843	C₂	semidirect
6. C₈×#11924031	C₂	semidirect
46. #11924031	M₁₆	semidirect
77. C₈×C₇₃	M₃₂	semidirect
107. C₇₃	#11409	semidirect
183. C₈	#14556665	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22626288	2
C₄×#14556665	2
#22624846	2
#22624844	2
#22624843	2
C₈×#11924031	2
M₁₆×#10596831	2
#11409	73